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September 29, 2025Open Access

C*-like modules and matrix p-operator norms

Key Points

C*-like pairs can maintain Holder duality under certain algebra-valued conditions, enhancing understanding of operator norms.
Our findings demonstrate that direct sums of the C*-like module remain C*-like when using block diagonal subalgebras.
We present counterexamples illustrating breakdowns of this property in non-block diagonal cases, emphasizing limitations.
The study's insights contribute to ongoing research in functional analysis and operator theory with potential applications.

Abstract

We present a generalization of H\"older duality to algebra-valued pairings via Lᵖ-modules. H\"older duality states that if p (1, ) and p^ are conjugate exponents, then the dual space of Lᵖ () is isometrically isomorphic to L^p^{} (). In this work we study certain pairs (Y, X), as generalizations of the pair (L^p^{} (), Lᵖ () ), that have an Lᵖ-operator algebra valued pairing Y X A. When the A-valued version of H\"older duality still holds, we say that (Y, X) is C*-like. We show that finite and countable direct sums of the C*-like module (A, A) are still C*-like when A is any block diagonal subalgebra of d d matrices. We provide counterexamples when A Mdᵖ (C) is not block diagonal.

C*-like modules and matrix p-operator norms

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Abstract

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